Flat ZipperUnfolding Pairs for Platonic Solids
Abstract
We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipperrefold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "ziprigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.
 Publication:

arXiv eprints
 Pub Date:
 October 2010
 arXiv:
 arXiv:1010.2450
 Bibcode:
 2010arXiv1010.2450O
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Discrete Mathematics;
 52B10;
 F.2.2
 EPrint:
 15 pages, 14 figures, 8 references. v2: Added one new figure. v3: Replaced Fig. 13 to remove a duplicate unfolding, reducing from 21 to 20 the distinct unfoldings. v4: Replaced Fig. 13 again, 18 distinct unfoldings